Nonstandard local empirical processes indexed by sets
Let \(X_ 1, X_ 2, \dots\) be independent random variables with common distribution \(\mu\) in \(R^ d\). Let \({\mathcal B}\) be the class of Borel subsets of \({\mathcal R}^ d\). Define the empirical measure indexed by \(B\) as \[ \mu_ n (B) = n^{-1} \sum^ n_{i = 1} I(X_ i \in B), \quad B \in {\mathcal B}, \] where \(I\) denotes the indicator function. Let \(\vec t \in \Sigma\) be fixed, where \(\Sigma\) denotes the support of \(\mu\). Let \({\mathcal C}\) be a class of Borel subsets of compact subset \(\Gamma\) of \({\mathcal R}^ d\) and \(\{k(n) : n \geq 1\}\) a sequence of positive constants such that \[ (\log \log n)^{-1} k(n) \to c \in (0, \infty) \quad (n \to \infty). \] Define the local empirical measure at \(\vec t\) indexed by \({\mathcal C}\) as \[ \theta_ n (C) = nk^{-1} (n) \mu_ n \biggl( \vec t + \bigl( n^{-1} k(n) \bigr)^{1/d} C \biggl) \quad (C \in {\mathcal C}). \] Let \(B({\mathcal C})\) be the class of bounded functions defined on \({\mathcal C}\), endowed with the topology of uniform convergence of \({\mathcal C}\). Under suitable assumptions on \(\mu\) and \(C\) the authors obtain the functional law of the iterated logarithm for \(\{\theta_ n (C), C \in {\mathcal C}\}\), i.e. the sequence is relatively compact in \(B({\mathcal C})\) with the limit set equal to \(\{g \in B ({\mathcal C}) : g (C) = \int_ C \varphi (\vec s) f(\vec t) d \lambda (\vec s) \leq 1\}\) with \(\int_{R^ d} ch(\varphi (\vec s)) f(\vec t)d \lambda (\vec s) \leq 1\), provided \(f(\vec t) > 0\), where \(h(x) = x \log x - x + 1\) \((x > 0)\); = 1 \((x = 0)\); \(= \infty\) \((x < 0)\) and \(\lambda\) denotes the Lebesgue measure on \(R^ d\).
functional laws of the iterated logarithm
scientific article; zbMATH DE number 758074